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Теория: Умножение больших натуральных чисел

Задание

Найти произведение чисел:

    \(\displaystyle 5\)\(\displaystyle 8\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 6\)\(\displaystyle 4\)\(\displaystyle 7\)

   
\(\displaystyle +\)        
   
\(\displaystyle +\)        
   

 

 

Решение

1. Первое действие.

Умножаем единицы числа \(\displaystyle 64{\underline 7}\) (это число \(\displaystyle 7\)) на число \(\displaystyle 58674{\small:}\)

 

\(\displaystyle 58674 \cdot 7=410718{\small.}\)

 

    \(\displaystyle 5\)\(\displaystyle 8\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 6\)\(\displaystyle 4\)\(\displaystyle \bf7\)

   \(\displaystyle \bf4\)\(\displaystyle \bf1\)\(\displaystyle \bf0\)\(\displaystyle \bf7\)\(\displaystyle \bf1\)\(\displaystyle \bf8\)
\(\displaystyle +\)        
  \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\) 
\(\displaystyle +\)        
 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)  

 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)

 

 

2. Второе действие.

Умножаем десятки числа \(\displaystyle 6{\underline 4}7\) (это число \(\displaystyle 4\)) на число \(\displaystyle 58674{\small:}\)

 

\(\displaystyle 58674 \cdot 4=234696{\small.}\)

 

Записываем результат так, чтобы единицы числа \(\displaystyle 23469{\underline 6}\) шли под той цифрой, на которую мы умножали (в нашем случае это десятки числа \(\displaystyle 6{\underline 4}7{\small,}\) то есть число \(\displaystyle 4\)).

 

    \(\displaystyle 5\)\(\displaystyle 8\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 6\)\(\displaystyle \bf4\)\(\displaystyle 7\)

   \(\displaystyle 4\)\(\displaystyle 1\)\(\displaystyle 0\)\(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 8\)
\(\displaystyle +\)        
  \(\displaystyle \bf2\)\(\displaystyle \bf3\)\(\displaystyle \bf4\)\(\displaystyle \bf6\)\(\displaystyle \bf9\)\(\displaystyle \bf6\) 
\(\displaystyle +\)        
 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)  

 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)

 

3. Третье действие.

Умножаем сотни числа \(\displaystyle {\underline 6}47\) (это число \(\displaystyle 6\)) на число \(\displaystyle 58674{\small:}\)

 

\(\displaystyle 58674 \cdot 6=352044{\small.}\)

 

Записываем результат так, чтобы единицы \(\displaystyle 35204{\bf 4}\) шли под той цифрой, на которую мы умножали (в нашем случае это сотни числа \(\displaystyle {\underline 6}47{\small,}\) то есть число \(\displaystyle 6\)).

 

    \(\displaystyle 5\)\(\displaystyle 8\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle \bf6\)\(\displaystyle 4\)\(\displaystyle 7\)

   \(\displaystyle 4\)\(\displaystyle 1\)\(\displaystyle 0\)\(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 8\)
\(\displaystyle +\)        
  \(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 4\)\(\displaystyle 6\)\(\displaystyle 9\)\(\displaystyle 6\) 
\(\displaystyle +\)        
 \(\displaystyle \bf3\)\(\displaystyle \bf5\)\(\displaystyle \bf2\)\(\displaystyle \bf0\)\(\displaystyle \bf4\)\(\displaystyle \bf4\)  

 \(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)\(\displaystyle {\small?}\)

 

 

4. Четвертое действие.

Складываем получившиеся числа:

 

   \(\displaystyle 4\)\(\displaystyle 1\)\(\displaystyle 0\)\(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 8\)
\(\displaystyle +\)        
  \(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 4\)\(\displaystyle 6\)\(\displaystyle 9\)\(\displaystyle 6\) 
\(\displaystyle +\)        
 \(\displaystyle 3\)\(\displaystyle 5\)\(\displaystyle 2\)\(\displaystyle 0\)\(\displaystyle 4\)\(\displaystyle 4\)  

 \(\displaystyle \bf3\)\(\displaystyle \bf7\)\(\displaystyle \bf9\)\(\displaystyle \bf6\)\(\displaystyle \bf2\)\(\displaystyle \bf0\)\(\displaystyle \bf7\)\(\displaystyle \bf8\)

 

В итоге имеем:

    \(\displaystyle 5\)\(\displaystyle 8\)\(\displaystyle 6\)\(\displaystyle 7\)\(\displaystyle 4\)
   x     
      \(\displaystyle 6\)\(\displaystyle 4\)\(\displaystyle 7\)

   \(\displaystyle 4\)\(\displaystyle 1\)\(\displaystyle 0\)\(\displaystyle 7\)\(\displaystyle 1\)\(\displaystyle 8\)
\(\displaystyle +\)        
  \(\displaystyle 2\)\(\displaystyle 3\)\(\displaystyle 4\)\(\displaystyle 6\)\(\displaystyle 9\)\(\displaystyle 6\) 
\(\displaystyle +\)        
 \(\displaystyle 3\)\(\displaystyle 5\)\(\displaystyle 2\)\(\displaystyle 0\)\(\displaystyle 4\)\(\displaystyle 4\)  

 \(\displaystyle \bf3\)\(\displaystyle \bf7\)\(\displaystyle \bf9\)\(\displaystyle \bf6\)\(\displaystyle \bf2\)\(\displaystyle \bf0\)\(\displaystyle \bf7\)\(\displaystyle \bf8\)

 

Ответ: \(\displaystyle 37962078{\small.}\)